Strong proximinality and polyhedral spaces

In any dual space X, the set QP of quasi-polyhedral points is contained in the set SSD of points of strong subdifferentiability of the norm which is itself contained in the set NA of norm attaining functionals. We show that NA and SSD coincide if and only if every proximinal hyperplane of X is strongly proximinal, and that if QP and NA coincide then every finite codimensional proximinal subspace of X is strongly proximinal. Natural examples and applications are provided

Various notions of best approximation property in spaces of Bochner integrable functions

We show that a separable proximinal subspace of X , say Y is strongly proximinal (strongly ball proximinal) if and only if L p ( I, Y ) is strongly proximinal (strongly ball proximinal) in L p ( I, X ), for 1 ≤ p < ∞ . The p = ∞ case requires a stronger assumption, that of ’uniform proximinality’. Further, we show that Y is ball proximinal in X if and only if L p ( I, Y ) is ball proximinal in L p ( I, X ) for 1 ≤ p ≤ ∞ . We develop the notion of ’uniform proximinality’ of a closed convex set in a Banach space, rectifying one that was defined in a recent paper by P.-K Lin et al. [J. Approx. Theory 183 (2014), 72–81]. We also provide several examples viz. any U -subspace of a Banach space has this property. Recall the notion of 3 . 2 .I.P. by Joram Lindenstrauss, a Banach space X is said to have 3 . 2 .I.P. if any three closed balls which are pairwise intersecting actually intersect in X . It is proved the closed unit ball B X of a space with 3 . 2 .I.P and closed unit ball of any M-ideal of a space with 3 . 2 .I.P. are uniformly proximinal. A new class of examples are given having this property

On strong proximinality in normed linear spaces

The paper deals with strong proximinality in normed linear spaces. It is proved that in  a compactly locally uniformly rotund Banach space, proximinality, strong proximinality, weak approximative compactness and  approximative compactness are all equivalent for closed convex sets. How strong proximinality can be transmitted to and from quotient spaces has also been discussed

Best approximation properties in spaces of measurable functions

We research proximinality of $\mu$-sequentially compact sets and $\mu$-compact sets in measurable function spaces. Next we show a correspondence between the Kadec-Klee property for convergence in measure and $\mu$-compactness of the sets in Banach function spaces. Also the property $S$ is investigated in Fr\'echet spaces and employed to provide the Kadec-Klee property for local convergence in measure. We discuss complete criteria for continuity of metric projection in Fr\'echet spaces with respect to the Hausdorff distance. Finally, we present the necessary and sufficient condition for continuous metric selection onto a one-dimensional subspace in sequence Lorentz spaces $d(w,1)$.Comment: 26 page